Sorting Cancer Karyotypes Sorting Cancer Karyotypes by Elementary Operationsby Elementary Operations

Michal Ozery-Flato and Ron ShamirSchool of Computer Science,

Tel Aviv University

Outline

• Introduction• Modeling the evolution of cancer karyotypes• The karyotype sorting problem• Combinatorial Analysis• Results

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Introduction

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http://www.ncbi.nlm.nih.gov/sky/skyweb.cgi

Normal female

karyotype

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The "Philadelphia chromosome"

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http://www.ncbi.nlm.nih.gov/sky/skyweb.cgi

Breast cancer

karytype (MCF-7)

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Chromosomal Instability

• A phenotype of most cancer cells.– Losses or gains of chromosomes result from

errors during mitosis

– Chromosome rearrangements are associated with "double strand breaks"

multi-polar mitoses

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Double Strand Breaks

• Constitute the most dangerous type of DNA damage– A successful repair ligates two matching

broken ends– Mis-repair can result in rearrangements (e.g.

translocations) or deletions

M.C. Escher, 1953

Double strand break

The Challenge

Analyze the evolution of aberration events in cancer karyotypes

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The Mitelman Database of Chromosome Aberrations in Cancer• Over 55,000 cancer karyotypes, culled from

over 8000 scientific publications• Can be parsed automatically (CyDAS parser

www.cydas.org)• The largest current data resource on cancer

genomes' organization

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Modeling the Evolution of Cancer Karyotypes

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The Normal Karyotype • Band = basic unit observable in karyotype. A

unique region in the genome, identified by integer• Normal Chromosome = interval of bands

– Two normal chromosomes are either disjoint or equivalent

• Normal karyotype = a collection of normal chromosomes– Usually contains two copies of each chromosome

(with the possible exception of the sex chromosomes)

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The Cancer karyotype

• Fragment = a sub-interval (>1 bands)

of a normal chromosome• Chromosome =

– One fragment, or a concatenation of several fragments

– Orientation-less: [1,4]::[37,40] [40,37]::[4,1]• Cancer karyotype = a collection of

chromosomes

concatenation (breakpoint)

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Elementary Operations

Breakage

Fusion

duplication

deletion

These operations can generate all known chromosomal aberrations!

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The Karyotype Sorting Problem

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The Karyotype Sorting (KS) Problem• Find a shortest sequence of elementary

operations that transforms the normal karyotype into given cancer karyotype

• Find the elementary distance = #operations in such a solution to KS.

???

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The Karyotype Sorting (KS) Problem(inverse formulation)

• Find a shortest sequence of inverse elementary operations that transforms the given cancer karyotype into the normal karyotype

???

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Inverse Elementary Operations

Breakage

Fusion

duplication

deletion

c-deletion

addition

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Assumptions

• ~95% of the karyotypes in the Mitelman Database have no recurrent breakpoints

• Assumptions:– The cancer karyotype contains no recurrent

breakpoints– Every added chromosome contains no

breakpoints

[20,39]::[12,1] Breakpoint ID={390,120}

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The Reduced Karyotype Sorting (RKS) Problem

• Assumptions reduced problem:– No breakpoints in the cancer karyotype

(i.e every chromosome is an interval)– No breakpoints created by fusions / additions All the normal chromosomes are identical

1 2 3 4 5 6 7 8 9 10 110 1 2 3 4 5 6 7 8 9 10 110The normal karyotype The cancer karyotype

breakage, fusion, c-deletion, addition

identical chromosomes

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Combinatorial Analysis

(RKS Problem)

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Extending the karyotypes

1 2 3 4 5 6 7 8 9 10 110The normal karyotype

1 2 3 4 5 6 7 8 9 10 110The cancer karyotype

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Parameter 1: f = #disjoint pairs of complementing interval ends

• Observation: f = -1 for fusion; f = 1 for breakage f {0,-1,-2} for c-deletion f {0,1,2} for addition

f =5

1 2 3 4 5 6 7 8 9 10 110

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The histogram

• Parameter 2: w = #bricks • Observations:

– w is even w = 0 for breakage / fusion w {0,2} for addition / c-deletion

1 2 3 4 5 6 7 8 9 10 110The cancer karyotype

1 2 3 4 5 6 7 8 9 10 110

The histogram

A wall with 2 bricks

A brick

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Simple Bricks• A brick is simple if

– no lower brick (in the same wall), and– no complementing interval ends

• Parameter 3: s = #simple bricks• Observation:

s {0,-1} for breakage s =0 for constrained-deletion– |s| 2 for addition

Simple bricks

1 2 3 4 5 6 7 8 9 10110

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The Weighted Bipartite Graph of Bricks

• Parameter 4: m = the minimum weight of a perfect matching

weight v-,v+:simple v-,v+:non-simple otherwisev- < v+ 2 0 1v+ < v- 0 2 1

1 2 3 4 5 6 7 8 9 10110

Positive bricks

Results

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Main Theorem

• The elementary-distance, d, satisfies:

w/2+f+s+m-2N d 3w/2+f+s+m-2N

N = #intervals in the normal karyotype

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Results (2)• Used the main theorem to devise a

polynomial-time 3-Approximation algorithm

• Combined with a greedy heuristic on real data (95% of Mitelman DB) optimal solutions computed for 100% of karyotypes– 99.99% cases : lower bound is achieved

(hence solution is optimal)– 30 cases: lower-bound+2 but actually optimal

(manual verification)

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Summary

• A new framework for analyzing chromosomal aberrations in cancer

• A 3-approximation algorithm when there are no recurrent breakpoints – 100% success on 57,252 karyotypes (with no

recurrent breakpoints) from the Mitelman DB.• Future work: handle recurrent breakpoints

– Analyze the remaining 5% of the karyotypes in the Mitelman DB.

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Thank for your attention.

Questions?